Abstract The BPS sectors of supersymmetric gauge theories have underlying algebraic structures, manifesting themselves in wall-crossing phenomena, BPS/CFT correspondences, Bethe/Gauge correspondences, twisted holographies, etc.  I will first explain how to construct BPS algebras for string theory on general toric Calabi-Yau threefolds, based on the crystal melting description of the BPS sectors. The resulting quiver Yangians, together with their trigonometric and elliptic versions, unify various known results and generalize them to a much larger class. Then I will discuss two applications of this construction: (1) the representations of these algebras can be described by subcrystals and give rise to new BPS counting problems; (2) these algebras can generate a large class of new Bethe/Gauge correspondences, in which the spin-chain is replaced by crystal-chains.